Problem: Find
\[\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 3 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 5 \\ 0 & 1 \end{pmatrix} \dotsm \begin{pmatrix} 1 & 99 \\ 0 & 1 \end{pmatrix}.\]
More generally,
\[\begin{pmatrix} 1 & a \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & b \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & a + b \\ 0 & 1 \end{pmatrix}.\]Therefore,
\[\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 3 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 5 \\ 0 & 1 \end{pmatrix} \dotsm \begin{pmatrix} 1 & 99 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 1 + 3 + 5 + \dots + 99 \\ 0 & 1 \end{pmatrix} = \boxed{\begin{pmatrix} 1 & 2500 \\ 0 & 1 \end{pmatrix}}.\]